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# behavior exists uniquely (finite case)

The following is a proof that behavior exists uniquely for any finite cyclic ring $R$.

###### Proof.

Let $n$ be the order of $R$ and $r$ be a generator of the additive group of $R$. Then there exists $a\in\mathbb{Z}$ with $r^{2}=ar$. Let $k=\gcd(a,n)$ and $b\in\mathbb{Z}$ with $a=bk$. Since $\gcd(b,n)=1$, there exists $c\in\mathbb{Z}$ with $bc\equiv 1\operatorname{mod}n$. Since $\gcd(c,n)=1$, $cr$ is a generator of the additive group of $R$. Since $(cr)^{2}=c^{2}r^{2}=c^{2}(ar)=c^{2}(bkr)=c(bc)(kr)=k(cr)$, it follows that $k$ is a behavior of $R$. Thus, existence of behavior has been proven.

Let $g$ and $h$ be behaviors of $R$. Then there exist generators $s$ and $t$ of the additive group of $R$ such that $s^{2}=gs$ and $t^{2}=ht$. Since $t$ is a generator of the additive group of $R$, there exists $w\in\mathbb{Z}$ with $\gcd(w,n)=1$ such that $t=ws$.

Note that $(hw)s=h(ws)=ht=t^{2}=(ws)^{2}=w^{2}s^{2}=w^{2}(gs)=(gw^{2})s$. Thus, $gw^{2}\equiv hw\operatorname{mod}n$. Recall that $\gcd(w,n)=1$. Therefore, $gw\equiv h\operatorname{mod}n$. Since $g$ and $h$ are both positive divisors of $n$ and $\gcd(w,n)=1$, it follows that $g=\gcd(g,n)=\gcd(gw,n)=\gcd(h,n)=h$. Thus, uniqueness of behavior has been proven. ∎

Note that it has also been shown that, if $R$ is a finite cyclic ring of order $n$, $r$ is a generator of the additive group of $R$, and $a\in\mathbb{Z}$ with $r^{2}=ar$, then the behavior of $R$ is $\gcd(a,n)$.

## Mathematics Subject Classification

16U99*no label found*13M05

*no label found*13A99

*no label found*

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